F := [x1/(1+t^2)+x2, -(x1+x2)*(1+1/(1+t^2)), -x3];
rho1 := [diff(F[1],x1),diff(F[2],x1),diff(F[3],x1)];
rho2 := [diff(F[1],x2),diff(F[2],x2),diff(F[3],x2)];
rho3 := [diff(F[1],x3),diff(F[2],x3),diff(F[3],x3)];
theta := Matrix([[rho2, -rho1, 0],[rho3, 0, -rho1],[0, rho3, -rho2]]);
with(LinearAlgebra):
with(linalg):

evalm(theta&*Matrix([[M1],[M2],[M3]]));

Find M1 M2 M3

Answer is
M1 = [-1, -1, 0]
M2 = [1, 0, 0]
M3 = [0, 0, -1]

Question 1. After googled, x*diff(F,t)+t*diff(F,x) for two variables, how about three variables or general ?

Question 2.

if deal with on variables case using the example in

http://demonstrations.wolfram.com/AFunctionInvariantUnderAGroupOfTransformations/

F := (x^2-x+1)^3/((x^2)*(x-1)^2)-y;
G:=y*diff(F,x)=0;
sol:=solve(G,y);

F(H)-F = 0
sol:=solve(F,x);
H := simplify(subs(y=F, sol[1]));

all above code can...

finding involutivedivision, autoreduction, involutivebasis for polynomial system,

which library contains these functions

Question 1:

Once using Phyics, i copy to notepad become the following, how to make it become normal syntax such as a+b*c

with(Physics):

Physics[`*`](Physics[`*`](Physics[`*`](Physics[`^`](v(t), 2)

ex := {v*t...

Question 2:
forget how to make some patterns in terms of x1, x2

for example would like to make repeated x/(x+1) to be replaced by x1

similar to

http://www.singular.uni-kl.de/Manual/html/sing_951.htm
http://www.singular.uni-kl.de/Manual/html/sing_753.htm

i would like to find all modules in the complex
for example

0 -> A -> B -> C -> 0

would like to find B and C if know A

which book describe this algorithm for polynomial system